Results obtained are summarized as follows: (1) A band-theoretic concept holds for stationary nonlinear localized modes in nonlinear lattices. Namely, stationary nonlinear localized modes can be interpreted as those eigenstates which are separated from the top, the bottom, or the both (duality) side of, the frequency or energy band of the corresponding linear lattices.by the intrinsic nonlinearity of the systems. (2) This gives us a hint on how to get approximate analytical results for moving nonlinear localized modes. (3) The concept appears general, suggesting the ubiquity of the existence of stationary nonlinear localized modes. (4) Except for possible existence of a critical value for the strength of nonlinearity, no basic difference exists between one-and higher dimensional cases for the appearance of the stationary nonlinear localizded modes. This suggests that in discrete lattices, no collapsing of stationary nonlinear modes exists. (5) Concrete results, both analytical and numerical, are given on the properties of intrinsically discrete nonlinear modes: (a) nonlinear localized modes in simple-cubic lattices (d = 3) (b) strongly localized modes in the diatomic Toda lattice and (c) duality of the existence, for a given nonlinearity, of stationary nonlinear localized modes in sine-lattices.
A poster will also be presented on a very recent result obtained in collaboration with M. Peyrard on a new kind of nonlinear localized modes, nonlinear rotating modes,, in coupled rotator systems (sine- lattices) exhibiting very strong localization and possessing intrinsically discrete character.